

where \(f\)  and \(g\)  are proper lower semicontinuous functions, \(Q\)  is a \(C^1\)  function with local Lipschitz continuous gradient \(\nabla Q\) , and \(X\) , \(Y\)  are the corresponding finite dimensional Hilbert spaces. Throughout this paper, we assume all the variables, the spaces, and the operators including the integrals setting are all finite dimensional. For the half quadratic models, (REF ) can be designed for the following equivalent minimization problem, where the pioneering work can be find in [1]}, [2]}:
\(\inf _{u \in X} F(u), \quad F(u):=D(u) + R(u) = \inf _{y \in Y}L(u,y).\) 
